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ABSTRACT. We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P 4. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P 4 with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components. 相似文献
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Density-functional theory is used to study the nuclear magnetic resonance (NMR) indirect nuclear spin-spin coupling constants in C60. Knowledge of these coupling constants may help in the analysis of future experimental NMR studies of 13C-enriched C60. At the Becke 3-parameter Lee-Yang-Parr (B3LYP) Kohn-Sham level, the one-bond couplings within pentagons and between pentagons are 62 Hz and 77 Hz, respectively; the corresponding geminal couplings are 7 Hz and 1 Hz, respectively. Except for the vicinal couplings (about 4 Hz), the long-range couplings are all 1 Hz or smaller. This is the largest theoretical calculation to date of the complete set of indirect nuclear spin-spin coupling constants of a molecular system; it has been made possible by solving the response equations only for the perturbing operators related to one nuclear magnetic moment, making the calculation feasible. 相似文献
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